Figure 1(a) shows in the Fourier space (momentum space) *k**x*-*k**z*-*k* of the physical space (position space) *x-z-t*, where *x* and *z* are the transverse and longitudinal space axes and *t* is the time axis, the ST spectrum of a ST wave packet lies in the intersecting curve between the light-cone and a spectral plane parallel to the *k**x*-axis [8, 9, 12–25], and satisfies

$$\cos \Delta \theta =\frac{{{\omega _0}+\Omega \cot \alpha }}{{{\omega _0}+\Omega }}$$

1

,

where, *Δθ* is the spectral-dependent propagating angle with respect to that of *ω**0*, *ω**0* is the vertex frequency of the ST spectrum defined as the intersecting point between the ST spectrum and the plane *k**x* = 0 at *k**z* > 0 and has the coordinates (*k**x*, *k**z*, *ω/c*) = (0, *k**0*, *ω**0**/c*), Ω = *ω* – *ω*0 is the frequency difference, and *α* is the tilt angle of the spectral plane with respect to the *k**z*-axis, which determines the velocity of a ST wave packet *v**g*/*c* = tan*α* [13, 23]. Figure 1(a) illustrates a superluminal and a subluminal ST wave packets with *α**1* > 45º and *α**2* < 45º at two different vertex frequencies, respectively.

Figure 1(b) shows in the type-I (o + o → e) collinear OPA, three waves propagate along the *z*-axis, and the energy is transferred from the strong pump (e-light) to the amplified signal (o-light) and the generated idler (o-light) in a thin nonlinear crystal (BBO here). We define *x-y-z* and *x’-y’-z’* are the lab and crystal coordinates, respectively; *θ**s*, *θ**i* and* θ**p* are propagating angles of the signal, idler and pump in the *x’-y’-z’* coordinates; because the pump always propagates along the *z*-axis, *Δθ**s* = *θ**s* - *θ**p* and *Δθ**i* = *θ**i* - *θ**p* are propagating angles of the signal and idler in the *x-y-z* coordinates. The relative rotation angle between the *x-y-z* and *x’-y’-z’* coordinates about the *y*/*y’*-axis is *θ**p* and changes the refractive index of the pump (e-light). Traditional collinear and non-collinear phase-matchings (momentum conservation) and energy conservation are illustrated in Fig. 1(c). If a ST wave packet can satisfy the amplification condition, Fig. 1(c) shows, apart from energy conversion, whose phase-matching should contain a collinear phase-matching for the vertex frequency *ω**0* and a series of spatially symmetric non-collinear phase-matchings for other frequencies *ω* = *ω**0* + Ω. Using a 515 nm pump, phase-matching curves for different pump angles *θ**p* from 23.02º to 23.42º every 0.1º are calculated and shown in Fig. 1(d). Here, we use solid and dash curves denote the signal and idler, respectively. The pump is always along *k**x* = 0, for the collinear phase-matching, both the signal and idler are along *k**x* = 0; and for non-collinear phase-matchings, the signal and idler are at either side of the pump. Comparing the angle-dependent phase-matching spectrum (here also named as ST spectrum) of the signal or idler [see Fig. 1(d)] with the ST spectrum of ST wave packets [see Fig. 1(a)], they have quite similar profiles in the *λ*-*k**x* (*ω*/*c*-*k**x*) plane, i.e., continuous symmetric conic curves about *k**x* = 0. If two curves overlap with each other, the vertex frequency *ω**0* and other frequencies *ω* = *ω**0* + Ω of a ST wave packet would be simultaneously amplified by collinear and non-collinear OPAs, respectively, generating an idler that is also a ST wave packet. The amplified signal and the generated idler would have different frequencies for the non-degenerate OPA and the same vertex frequency for the degenerate OPA, respectively.

To verify this judgment, we derived the analytical expression of the angle-dependent phase-matching spectrum (ST spectrum) of OPA under the narrowband approximation, which satisfies

$$\cos \Delta {\theta _s} \approx \frac{{{\omega _{s0}}+{\Omega _s}{{\left( {{\omega _{s0}}n_{s}^{2}+\frac{{{{\left( {{\omega _p}{n_p} - {\omega _{s0}}{n_s}} \right)}^2}}}{{{\omega _p} - {\omega _{s0}}}}} \right)} \mathord{\left/ {\vphantom {{\left( {{\omega _{s0}}n_{s}^{2}+\frac{{{{\left( {{\omega _p}{n_p} - {\omega _{s0}}{n_s}} \right)}^2}}}{{{\omega _p} - {\omega _{s0}}}}} \right)} {{\omega _p}{n_p}{n_s}}}} \right. \kern-0pt} {{\omega _p}{n_p}{n_s}}}}}{{{\omega _{s0}}+{\Omega _s}}}$$

2

,

where, *Δθ**s* is the propagating angle of the frequency *ω**s* = *ω**s0* + Ω*s* with respective to the vertex frequency *ω**s0* within the signal ST wave packet in the *x-y-z* coordinates, *n**s* and *n**p* are refractive indices of the vertex frequency *ω**s0* and the pump frequency *ω**p*, and constant *n**s* is used for all frequencies of the signal under the narrowband approximation. By comparing Eqs. (2) with (1), the ST spectrum (angle-dependent phase-matching spectrum) of the signal in OPA has the same expression with that of ST wave packets. This indicates that a ST wave packet can be perfectly amplified by a matched OPA. The tilt angle *α**s* of the spectral plane in the Fourier space is given by

$$\cot {\alpha _s} \approx \frac{{{\omega _{s0}}n_{s}^{2}+\frac{{{{\left( {{\omega _p}{n_p} - {\omega _{s0}}{n_s}} \right)}^2}}}{{{\omega _p} - {\omega _{s0}}}}}}{{{\omega _p}{n_p}{n_s}}}$$

3

.

The refractive indices *n**s* and *n**p* in the crystal can be calculated by the Sellmeier equation for *ω**s0* and *ω**p*, respectively, and the pump angle *θ**p* is contained in *n**p*. Because three-wave coupling equations show the signal and idler are symmetric in mathematics [see Eq. (5) in Methods], the generated idler is also a ST wave packet, whose ST spectrum *Δθ**i*(Ωi) and spectral plane tilt angle *α**i* are also described by Eqs. (2) and (3) by simply replacing the subscript *s* by *i*.

Solving the collinear phase-matching [Eq. (5) in Methods] in a 515 nm pumped type-I BBO crystal, at different pump angles *θ**p*, two phase-matched vertex wavelengths *λ**0* of the signal and idler in a double ST wave packet can be calculated, and by substituting into Eq. (3), their velocities are obtained (*v**g*/*c* = tan*α*) and shown in Fig. 2. A signal/idler and its idler/signal lie in red and black curves, respectively. When the pump angle is *θ**p* < 23.32º or *θ**p* = 23.32º, OPA is nondegenerate or degenerate, respectively, and the double ST wave packet has “different wavelengths and velocities” or “same wavelength and velocity”, respectively. In the nondegenerate OPA, velocities of a double ST wave packet are superluminal and subluminal when the vertex wavelengths are shorter and longer than the degenerate-wavelength (1030 nm here), respectively. In this case, the two ST wave packets will separate in time during propagation, which can also be separated in spectrum. In the degenerate OPA, the signal and idler have a same vertex wavelength of the degenerate-wavelength (1030 nm here), a same 45º tilt angle of spectral planes in the Fourier space [see Eq. (3)], and a same velocity of the light speed (as introduced in Refs. [13, 25], ST wave packets become luminal plane wave pulses). However, Fig. 1(d) shows only the vertex wavelength satisfies OPA, and the amplified signal and the generated idler become quasi-monochromatic (the degenerate-wavelength) luminal plane wave pulses.

In addition to the strict phase-matching (*Δk* = 0), small phase-mismatchings (*Δk* → 0) can also support OPAs. Red zones in Figs. 3(a) and 3(b) show gain spectra in a 2 mm thin type-I BBO crystal, when the pump wavelength, angle, and intensity are 515 nm, *θ**p* = 23º, and 10 GW/cm2, respectively. Although two gain spectra in both short- and long-wave ranges have some ST correlations [see Figs. 3(a) and 3(b)], Figs. 3(c) and 3(d) show the supported pulsed beams have Gaussian ST distribtuions, which accordingly are not diffraction-free. H. E. Kondakci et al. have defined a parameter of the correlation uncertainty *δλ* between *k**x* and *λ*, which is inversely proportional to the propagation-invariant length [8]. Large correlation uncertainties *δλ* = 43 and 99 nm of short- and long-wave gain spectra distort ST correlations, and that is why no diffraction-free beam has been previously reported in OPA, optical parametric generation (OPG), or optical parametric oscillation (OPO). A ST wave packet is optimized [using Eq. (3)] for seeding the above OPA, whose ST spectrum is shown by the yellow zone in Fig. 3(a), and parameters include a reduced correlation uncertainty *δλ* = 2 nm, a vertex wavelength *λ**0* = 855 nm, a bandwidth *Δλ* = 40 nm, and a tilt angle of the spectral plane in the Fourier space *α* = 45.0572º. The ST spectrum of the generated idler is illustrated by the yellow zone in Fig. 3(b), and parameters are a correlation uncertainty *δλ* = 4.7 nm, a vertex wavelength *λ**0* = 1295 nm, a bandwidth *Δλ* = 105 nm, and a tilt angle of the spectral plane in the Fourier space *α* = 44.9111º. The 10 GW/cm2 strong pump enables 400× gain for the signal and generates an idler of the same energy level. Figures 3(e) and 3(f) show the pulsed beams supported by the ST spectra of the signal and idler, and both have a typical distribution of ST wave packets with a butterfly-like shape and a central peak [8, 9]. Although the “central peaks” are enlarged in space and time [compare Figs. 3(e, f) and 3(c, d)], which do not spread out during long-distance propagation much exceeding the Rayleigh length. We can also find some small mis-matchings between ST spectra of signal and idler ST wave packets and strict phase-matching curves (dash lines) in Figs. 3(a) and 3(b), especially when wavelengths are away from the vertexes. It is because the narrowband approximation for Eq. (2) cannot be strictly satisfied. However, ST spectra of signal and idler ST wave packets still lie in gain spectra of OPA, which doesn’t affect amplification. In experiments, a near-perfect phase-matching can be obtained by reducing both the bandwidth *Δλ* and the correlation uncertainty *δλ* of the signal ST wave packet.

The free propagation of the amplified signal and the generated idler in vacuum are shown in Fig. 4. OPA happens at the position *z* = 0, where the signal and idler overlap in space and time. After *z* = 0, the separation between two ST wave packets increases with propagation due to different velocities of the signal and idler *v**gs* = 1.0020*c* and *v**gi* = 0.9969*c*. Figure 4 also shows the propagation-invariant length of the idler is shorter than that of the signal. The first reason is the Rayleigh length for a fixed beam waist *w**0* decreases with increasing the wavelength (e.g., Gaussian beam *Z**R* = *πw**0**2**/λ*). Here, both central-peak waists of signal and idler ST wave packets are around *w**0* = 23.5 µm, however due to a longer wavelength (1295 and 855 nm vertex wavelengths for the idler and signal), the Rayleigh length of the idler *Z**Ri* = 1.3 mm is shorter than that of the signal *Z**Rs* = 2.0 mm. The second reason is the correlation uncertainty of the idler *δλ* = 4.7 is larger than that of the signal *δλ* = 2 and accordingly shortens the propagation-invariant length. Even so, Fig. 4 shows both the propagation-invariant lengths have far exceeded their Rayleigh lengths of *Z**Rs* = 2.0 mm and *Z**Ri* = 1.3 mm.

In summary, we have proposed a principle of amplifying a ST wave packet to high-energy by the type-I collinear OPA, where an idler ST wave packet would be generated simultaneously. Our numerical simulation verified the feasibility of this principle and demonstrated an as yet unreported strong double ST wave packet. In the nondegenerate OPA, two collinear-propagating ST wave packets at short and long wavelengths (compared with the degenerate-wavelength) are superluminal and subluminal, respectively, which accordingly separate in time during propagation. In the degenerate OPA, the double ST wave packet becomes a quasi-monochromatic (the degenerate-wavelength) luminal plane wave pulse. Amplifying ST wave packets in energy will introduce this new type light source to strong-field optics for studying ST-light–matter interactions. Moreover, simultaneously generated idlers can turn ST wave packets to other wavelengths, such as infrared, visible and ultraviolet wavelengths, and switch velocities between superluminal and subluminal. This discovery will expand the range of applications of ST wave packets.